No, it's not "period". Because some people might claim, and have arguments, that including games with strong action elements is as bad as making Castlevania an RPG. That's the whole problem.
What are these "arguments" you mention?
There is no clear cut distinction. For example in Gothic there are people who, to my knowledge, can defeat very powerful opponent by skill alone, using their own reflexes, regardless of their char low stats. One could claim that it makes it an action game with only elements of a crpg. But even if we agree that this particular game is a crpg despite all that, it's still not clear cut at all as you claim it to be.
You said it yourself:
"People who", not "the overwhelming majority". The overwhelming majority will find a game whose combat makes it almost impossible to beat the game without improving their character's skill and equipment. In other words, character and equipment progression in Gothic is anything but trivial.
Different kinds of games that are often referred to as RPG:
a wizardry-like blobber,
a party based, combat focused, isometric crpg,
a fpp action/rpg hybrid,
a C&C-centered "storyfag" rpg.
Some of those often don't even feel like they belong to same genre despite having evolved from the same source.
I mean, can you blame the genre? It has evolved for a long time already. Some have suggested Wizardry is not even an RPG. I personally believe it is one, but "dungeon crawler" fits it better.
And yet you postulate that the term "crpg" should encompass all of them. From my perspective, it makes little sense.
It makes perfect sense when I stop analyzing the mechanics and start analyzing the experience I get from them.
- Wizardry: I feel like I am the party, with strengths, weaknesses, and limitations.
- Baldur's Gate: likewise.
- New Vegas & Deus Ex: likewise, with the caveat you just control one character (like in Fallout).
- Planescape: Torment: likewise. Though I'm guessing you meant to say Alpha Protocol, which I haven't played.
With a precise enough definition when a certain game is said to belong to genre X, we can know what game to expect in general terms. Makes it easier to buy/research games.
The issue is that you and the elf are trying to define an RPG by its mechanics and not by the overall experience. Two games with the same mechanics may offer completely different experiences, to the point one is clearly an RPG (and one of the greatest ever made), while the other is considered a post-apocalyptic walking simulator by comparison, where all you do is shoot stuff until it dies. This is why New Vegas is an RPG, and people claim Fallout 4 is an FPS: because while the elements are there, the game simply doesn't make use of them, and prefers to play base building and Call of Duty.
New Vegas doesn't offer the experience of your typical action game. It doesn't offer the experience of your typical shooter either. In the same vein, Castlevania doesn't offer the experience of your typical RPG, but it does offer the experience of your typical action platformer "Metroidvania". And that's exactly what it is: a Metroidvania with RPG elements tacked on for novelty.
Some users say "it's an RPG if I like it!", but despite their sarcastic attitude I do believe there's some truth to it. I simply wouldn't enjoy New Vegas if it was just a Quake clone set in the Fallout universe. Why? Because it would lose everything that makes it an RPG and not just a bad shooter (bad gunplay and very few enemies to kill + uninteresting scenarios to kill them in).